Thermal Control of Plasmonic Surface Lattice Resonances

Plasmonic metasurfaces exhibiting collective responses known as surface lattice resonances (SLRs) show potential for realizing flat photonic components for wavelength-selective processes, including lasing and optical nonlinearities. However, postfabrication tuning of SLRs remains challenging, limiting the applicability of SLR-based components. Here, we demonstrate how the properties of high quality factor SLRs are easily modified by breaking the symmetry of the nanoparticle surroundings. We break the symmetry by changing the refractive index of the overlying immersion oil by controlling the ambient temperature of the device. We show that a modest temperature change of 10 °C can increase the quality factor of the SLR from 400 to 750. Our results demonstrate accurate and reversible modification of the properties of the investigated SLRs, paving the way toward tunable SLR-based photonic devices. More generally, we show how symmetry breaking of the environment can be utilized for efficient and potentially ultrafast modification of the SLR properties.

leaving only the nanoparticles on the glass substrate. The sample was then rinsed with IPA and dried with nitrogen.
Before the measurements, we covered the metasurface with index-matching oil and an anti-reflection (AR) coated coverslip with the AR wavelength band at 1000-1300 nm. This way, the nanoparticles were assured to have a homogeneous surrounding, and we avoided any Fabry-Pérot resonances resulting from multiple reflections from different interfaces present in the fabricated devices.

Localized Surface Plasmon Resonance
Polarizability for NPs of arbitraty shape is given by 1 where V is the volume of the nanoparticle (NP), ε m and ε s are the permittivities of the metal and surrounding media respectively, L i is a factor depending on the geometry of the particle, and index i marks the relevant dimension. While L i has been analytically solved for shapes such as slabs and ellipsoids, 1 a general analytical solution for complicated shapes, such as the V-shaped particles used in this work, does not exist. Consequently, we used a value of L y = 0.26, that was found by trial and error to match well with the experiments.
Equation (1) assumes the electric field to be static at a given time, which produces accurate results while the dimensions of the NPs are smaller than 1% of the incident wavelength. 2 With our NPs, however, a modified long-wavelength approximation (MLWA) using dynamic perturbations must be used for accurate modelling of the localized surface plasmon resonances (LSPRs). After applying MLWA, polarizability can be written as 3 where α i is the static polarizability, and k is the wavenumber of the incident field. Figure 1 shows experimental data next to analytical model using equations (1) and (2).
The experimental data is measured from the same metasurface as the SLRs in this work. It should be noted that the LSPR is not pure due to the presence of the second order diffractionmode near 550 nm. In the equation (1), experimental data for aluminum permittivity from Rakić 4 was used for ε m . The surroundings were modelled using the permittivity of the SCHOTT -multiple purpose D 263® T eco Thin glass. In the Equation (2), a NP dimension of a y = 75 nm was used, corresponding approximately to half of the length of the NP in y-direction. In Equation (1) a geometrical factor a of L y = 0.26 was used.

Lattice Sum Approach
Lattice sum approach (LSA) method was used to support the experimental results with semianalytical calculations. LSA is a simplified version of the discrete-dipole approximation (DDA) method that allows to calculate the response of optically coupled nanoparticles. An interested reader is referred to Ref. [5] for further details regarding the similarities and differences between the LSA and the DDA approaches. Shortly put, in LSA the effects of coupling between nanoparticles are reduced to a single constant for all nanoparticles. The reduction is done with the loss of generality in non-infinite lattices, with LSA being unable to model dipole moments of the nanoparticles near the edges of the lattice. The LSA is based on several assumptions. Most importantly, all the nanoparticles are treated as pointlike objects which scatter the incoming light as electric dipoles. In LSA specifically, it is assumed that all the nanoparticles are identical, and experience identical environments with respect to each other.
In LSA, a concept of effective polarizability α * is introduced, which associates the effectively induced dipole moments p to the incident electric field E via p = ε 0 ε s α * E, where ε 0 and ε s are the vacuum permittivity and relative permittivity of the surroundings, respectively. The effective polarizability is given by where i denotes a component of the quantity in Cartesian direction, α i is the single particle polarizability, and S i is the lattice sum.
The lattice sum in equation (3) can be expressed in a homogeneous environment as 5 where N is the number of particles taken into account, r j is the distance between j th particle and the center particle, and φ i,j is the angle between the i th dipole moment component and the vector from the center particle to j th particle. The lattice sum S can be derived from DDA by assuming identical dipole moments for all NPs. 5 In this work, the studied NP arrays lied on an interface between Olympos IMMOIL-F30CC -immersion oil and SCHOTT -multiple purpose D 263® T eco Thin Glass. This caused slight heterogeneity in the surroundings of the SLR at room temperature. To model the heterogeneity, the coupling of NPs was thought to develop independently in the different materials. We assumed the heterogeneous lattice sum to be a weighted mean of two different homogeneous lattice sums arising in the different surrounding materials. Changing the weights make the peak location to shift continuously between the two homogeneous peak locations. Therefore, the weights could be determined by comparing the shift in the peak location in the LSA calculations to the ones extracted from experimental results. The analysis arrived at even weights for both materials. Therefore the heterogeneous lattice sum is calculated in our case as follows where k oil and k glass are the wavenumbers in the (Olympos IMMOIL-F30CC -immersion) oil and (SCHOTT -multiple purpose D 263® T eco Thin) Glass respectively, given by where λ is the wavelength, and n oil (λ, T ) and n glass (λ, T ) are the refractive indices of the oil and glass respectively. Dispersion data for n glass (λ, T ) was taken from. 6

Effects of temperature on the permittivity of the metal
It is difficult to model the temperature dependence of the aluminum permittivity using the Drude-model near visible and infra-red frequencies due to interband excitations near 800 nm. However, since the experiments were carried out near the wavelength of 1100 nm, the interband excitations taking place near 800 nm can be expected to play a negligible role.
Nevertheless, we decided to estimate the temperature dependence of the permittivity of metallic NPs, and subsequent changes in the SLRs of periodic nanoparticle arrays, for gold NPs. This approach was chosen because the permittivity of aluminum can be expected to behave very similarly to that of gold near the wavelength of 1100 nm. The permittivity of gold was modelled using the Drude model: 10 where ε m is the permittivity of gold, ε ∞ is the high-frequency permittivity, ω p is the plasma frequency of the metal, ω is the angular frequency of the incident light, and γ is the relaxation constant for the free electrons. Plasma frequency is given by 10 where n 0 is the free electron number density in the metal, e is the elementary charge, ε 0 is the vacuum permittivity, m is the mass of electron, β is the linear thermal expansion coefficient and ∆T = T − T 0 is the difference to the temperature of comparison T 0 . The term 1 + 3β∆T arises from the thermal expansion of the material, which is the temperature dependent effect affecting the plasma frequency.
The thermal relaxation constant γ is comprised of electron-electron contribution γ e−e (T ) and electron-phonon contribution γ e−ph (T ), which are given by 10 where the factor A is dependent on the properties of the conduction band of the metal, It is evident that the slight temperature dependence of the permittivity of the metal has little effect on the SLR in the small temperature range (∆T = 30 K) we used in our work.
Attempts to model the effects of the temperature dependence of aluminum were therefore omitted. However, we note that in the case of perfectly homogeneous environment, the intrinsic temperature dependence of the metal could significant change the formation of the SLR, when broad enough temperature ranges are used. In particular, looking at our results, it is evident that the Q-factors of the SLRs could be increased when samples would be cooled to cryogenic temperatures. Other effect affecting SLRs in homogeneous environments is the shifting of the spectral location of the SLR with temperature dependent refractive index of the homogeneous substrate.

LSA data
All Q-factors, peak locations λ c and extinctions were determined in this work by fitting a Lorentzian line shape to the extinction spectra. Lorentzian is a line shape given by equation where λ is the wavelength, I is the peak extinction of the resonance peak, and γ 0 is the half width at half maximum. The Q-factor is therefore given by The fitted models are shown along with the experimental data on Figure 3.